Making the Long Code Shorter
2012 (English)In: Foundations of Computer Science (FOCS), 2012 IEEE 53rd Annual Symposium on, IEEE Computer Society, 2012, 370-379 p.Conference paper (Refereed)
The long code is a central tool in hardness of approximation, especially in questions related to the unique games conjecture. We construct a new code that is exponentially more efficient, but can still be used in many of these applications. Using the new code we obtain exponential improvements over several known results, including the following: 1) For any ε > 0, we show the existence of an n vertex graph G where every set of o(n) vertices has expansion 1-ε, but G's adjacency matrix has more than exp(logδ n) eigenvalues larger than 1 - ε, where δ depends only on ε. This answers an open question of Arora, Barak and Steurer (FOCS 2010) who asked whether one can improve over the noise graph on the Boolean hypercube that has poly(log n) such eigenvalues. 2) A gadget that reduces unique games instances with linear constraints modulo K into instances with alphabet k with a blowup of Kpolylog(K), improving over the previously known gadget with blowup of 2Ω(K). 3) An n variable integrality gap for Unique Games that survives exp(poly(log log n)) rounds of the SDP + Sherali Adams hierarchy, improving on the previously known bound of poly(log log n). We show a connection between the local testability of linear codes and small set expansion in certain related Cayley graphs, and use this connection to derandomize the noise graph on the Boolean hypercube.
Place, publisher, year, edition, pages
IEEE Computer Society, 2012. 370-379 p.
, Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS, ISSN 1523-8288
Locally Testable Codes, Long Code, Small set expansion, Unique games conjecture
IdentifiersURN: urn:nbn:se:kth:diva-112850DOI: 10.1109/FOCS.2012.83ISI: 000316999700040ScopusID: 2-s2.0-84871951816OAI: oai:DiVA.org:kth-112850DiVA: diva2:587548
53rd Annual IEEE Symposium on Foundations of Computer Science, FOCS 2012; New Brunswick, NJ; 20 October 2012 through 23 October 2012
QC 201301162013-01-142013-01-142013-01-16Bibliographically approved